Old Nuts

In turn 4 people throw away three nuts from a pile and hide a quarter of the remainder finally leaving a multiple of 4 nuts. How many nuts were at the start?

Sixational

The nth term of a sequence is given by the formula n^3 + 11n . Find the first four terms of the sequence given by this formula and the first term of the sequence which is bigger than one million. Prove that all terms of the sequence are divisible by 6.

Modular Knights

Try to move the knight to visit each square once and return to the starting point on this unusual chessboard.

Factorial Fun

Stage: 5 Challenge Level:

(a) We denote the product of the first 20 natural numbers by 20! and call this 20 factorial.

What is the highest power of 5 which is a divisor of 20 factorial? Just how many factors does 20! have altogether?

b) Show that the highest power of $k$ that divides 500!, where $k$ is an integer and $k^{(t + 1)} > 500 > k^t$ is ,

$$[500/k] + [500/k^2] + ... + [500/k^t].$$

where the square brackets are used to denote the integer part of the number inside.

(c) How many factors does $n!$ have?